help with this calc problem plz?

Question

Let F(x) = the integral of (t^2 + t)^.5 from 1 to 2x.

(a) Find F'(x)

(b) Find the domain of F.

(c) Find the limit of F(x) as x approaches 1/2

(d) Find the length of the curve y = F(x) for [1 <(=) x <(=) 2]

Answer 1

(a) derivation of an integral with upper limit variable is the function under integral times derivation of the uper limit. in your case
F'(x) = 2*sqrt(x^2 + x).

(b) Domain of F(x) are the values of x for which F(x) exists (is real). For values -1
(c) for x = 1/2 lower and upper integral limits are the same therefore F(1/2) = 0.

(d) If F(x) is the curve then small piece of the curve dl = sqrt(dx^2 + (F'(x)*dx)^2) = |using (a)| = dx*sqrt(1 + 4*x^2 + 4*x) = (2x+1)*dx. The length is the integral between x=1 and x=2 of (2x+1)dx = (x^2+x) between 1 and 2 = 4 + 2 - 1 - 1 = 4.